Abstract
This essay shows that large parts of fuzzy set theory are actually subfields of sheaf theory, respectively, of the theory of complete Ω -valued sets. Hence fuzzy set theory is closer to the mainstream in mathematics than many people would expect. Part I of this essay divided into a series of two papers presents such basic concepts as Ω -valued equalities, espaces étalés, singleton monad, the change of base and the subobject classifier axiom. The application of these tools to the sheaf-theoretic foundations of fuzzy sets will appear in Part II.
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